Re: Question
- From: magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin)
- Date: Thu, 27 Apr 2006 19:37:25 +0000 (UTC)
In article <1146165828.532400.65880@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
zuhair <zaljohar@xxxxxxxxx> wrote:
Arturo Magidin wrote:
[.snip.]
It doesn't differentiate
between
0.9999........
which contains Omega of nines in it after the decimal point.
and 0.9999....... 9 which contains Omega+1 of nines in it after the
decimal point.
That expression does NOT represent a sequence, so it is not under
consideration. It is merely a symbolic string of symbols which does
not represent a real number.
???????????? this is an escape rather than an answer.
Just because you don't like the answer does not mean it is not an
answer. Your question was the logical equivalent of "what kind of duck
is a fox terrier?" It would seem you would consider the answer "a fox
terrier is not a duck" to be "an escape rather than an answer."
A sequence is, BY DEFINITION, a map whose domain is the natural
numbers. A decimal expansion represents a sequence. Therefore, a
decimal expansion can only have digits at positions that correspond to
natural numbers. When one writes a decimal expansion, in the world
everyone but you plays in, one is using this expansion as shorthand
for a specific equivalence class of Cauchy sequences of rationals
(under the standard definition of equivalence, not the one you think
"should" be the definition). Thus, decimal expansions, BY DEFINITION,
contain digits only in locations correspoding to natural
numbers. Omega is not a natural number. You can define a function from
omega + 1 = omega \/ {omega}, whose range are digits, but in that case
that function is quite simply not a sequence, just as a fox terrier is
not a duck. So the symbolic string of digits which represents the
function from omega\/{omega} which has value 9 at all elements is NOT
a sequence, and not a real number.
It is not an escape. It is a precise, logical, correct answer. Don't
like it? Too bad.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx
.
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